346 XV NORMED ALGEBRAS AND SPECTRAL THEORY Let s\~*U(s) be a representation of A in a Hilbert space H. A vector subspace E of H is said to be stable with respect to this representation if U(s)E c E for all s e A. If E is stable with respect to £/, then so is the closure E of E (3.11.4). If E is closed and if E' is the orthogonal supplement of E in H (6.3), then E' is also stable with respect to U. For if x e E and x' E E', we have (x \ U(s) • x') = ((U(s))*x \ x') = (U(s*)x | x') = 0 by hypo- thesis, hence U(s) - x' is orthogonal to all x e E and therefore belongs to E'. If E/iCs1) and U2(s) are the restrictions of U(s) to E and E', respectively, then the representation U is the Hilbert sum of Ut and U2 - (15.5.3) For a closed subspace EofH to be stable with respect to U, it is necessary and sufficient that PE U(s) = U(s)PE for all seA, where PE is the orthogonal projection on E (6.3). The condition is necessary, for if x e E we have PE - x = x and PE * (U(s) - x) = U(s) - x, because E is stable with respect to U; and if x e E' we have PE - x = 0 and PE • (U(s) * x) = 0, because E' is also stable with respect to U. Conversely, if the given condition is satisfied, then U(s) - x = PE - U(s) - x e E for all x e E and all s e A. On a Hilbert space H, an orthogonal projector is by definition any con- tinuous operator on H which is an orthogonal projection onto a closed sub- space of H. The importance of such projectors is due to (15.5.3) and to their characterization in terms of the structure of algebra with involution of JSP(H): (15.5.3.1) A continuous operator P on a Hilbert space H is an orthogonal projector if and only if it is idempotent and hermit tan. The necessity of these conditions has already been proved (11.5). Con- versely, if P2 = P = P*, then (P • x\y - P ' y) = (x\P - y - P2 • y) = 0 for all x, y e H. Since P(H) is also the kernel of 1H — P, it is closed, and H is the Hilbert sum of P(H) and P ~'(()). Hence the result. Suppose that H is the Hilbert sum of an infinite sequence (Hn) of sub- spaces which are stable with respect to the representation U. Let Un(s) denote the restriction of U(s) to Hn, so that for each n the mapping s\-* Un(s) is a representation of A in Hn. By abuse of language, tlie representation U is said to be the Hilbert sum of the representations Un. For each s e A and each x = Z xn6 H, where xn e Hn for each «, we have U(s) • x = £ Un(s) - xn, and